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Equations of Motion for a mass on a circular track with a linear spring

  1. Sep 11, 2012 #1
    1. The problem statement, all variables and given/known data

    A collar of mass m slides without friction along a circular track of radius R as
    shown in THE ATTACHMENT. Attached to the collar is a linear spring with spring constant K and
    unstretched length zero. The spring is attached at the fixed point A located a distance
    2R from the center of the circle. Assume gravity acts down and determine (a) the differential equation of motion for the collar in terms the angle θ and (b) the reaction force exerted by the track on the collaras a function of the angle θ.


    2. Relevant equations

    The transport theorem

    F = ma


    3. The attempt at a solution

    I have worked through this problem and am not confident in my answer. A second look would be much appreciated. Here's my answer:

    I created reference frame E: {er, eθ, ez} at mass m such that it's position is

    r = Rer

    By using the transport theorem for velocity and acceleration as seen by the intertial frame I came up with:

    v(m/I) = 0 + θ(dot)R[eθ]
    a(m/I) = θ(double dot)R[eθ] - (θ(dot))^2R[er]

    Next the forces acting on my mass are gravity, normal, and linear spring

    ∴Ʃ F = mg[JI] + N[er] + K(3R-0)[-er + JI]

    where JI = sinθ[er] + cosθ[eθ]

    when I substitute JI into my ƩF equation, set equal to ma(m/I), and drop the unit vectors, I get these two equations:

    1. mgsinθ + N -3KR + 3KRsinθ = -m(θ(dot))^2R
    2. mgcosθ + 3KRcosθ = mθ(double dot)R

    For my equation of motion in terms of θ, I only need to use the second equation, correct? Where θ(double dot) = (mgcosθ + 3KRcosθ)/(mR)

    and for the normal force (N) I just use the first equation and move everything to one side for N = -mgsinθ +3KR - 3KRsinθ -m(θ(dot))^2R

    Can someone please take a look at this? I feel as though my spring force
    Fs = K(3R-0)[-er + JI] is not correct, but I don't know what else it should be. Any help would be appreciated.

    Thank you.
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     

    Attached Files:

  2. jcsd
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